Time-dependent Fourier Research Articles

A time-dependent Fourier grid Hamiltonian method. Formulation and application to the multiphoton dissociation of a diatomic molecule in intense laser field

A time-dependent Fourier grid Hamiltonian method is proposed for studying real-time quantum dynamics of simple systems. The method can work with an arbitrary number of grid points ( N). Convergence with respect to N is fast. Results of test calculations on the dynamics of dissociation of a diatomic species modelled by an appropriate Morse oscillator in a continuous high intensity non-resonant IR laser field are presented. The dissociation is predicted to be characterized by the presence of a definite threshold of laser intensity and an induction period preceeding the onset of dissociation.

Evolution of weakly nonlinear waves in a cylinder with a movable piston

Small-amplitude wave motion in an inert gas confined between a moving piston and a fixed cylinder endwall is studied using the unsteady Euler equations. The waves, generated by either initial disturbances or piston motion, reflect back and forth in the cylinder on the acoustic timescale. The accumulated effect of these waves controls the bulk variations of velocity and thermodynamic variables on the longer piston timescale. Perturbation methods, based on the small ratio of acoustic to piston time, are employed to formulate the gasdynamic problem. The application of multiple timescaling allows the gasdynamic wave field to be separated from the bulk response of the gas. The evolution of the wave phenomena, including nonlinear wave deformation and weak shock formation during the piston passage time, is described in terms of time-dependent Fourier series solutions, whose coefficients are computed from a truncated system of coupled nonlinear ordinary differential equations. The long-time asymptotic flow field after shock formation is sawtooth-like, in which case the Fourier modes become decoupled. A remarkably simple relation between the shock amplitude and piston velocity is discovered. It is demonstrated that (i) the wave amplitude and frequency strongly depend on the piston motion; (ii) shock waves can be damped in a significant way by internal dissipation; and (iii) the mathematical approach developed in this study possesses certain advantages over the more traditional method of characteristics.

Dynamic compression and weak shock formation in an inert gas due to fast piston acceleration

Unsteady gasdynamic concepts are used to model the piston-driven compression of a confined gas. Perturbation methods, based on the limit of small piston Mach number, are used to construct solutions. The piston Mach number increases smoothly from zero to a maximum value, Mp = O(10−2) during an acoustic time period ta* = O(10−4 s). A linear a coustic field is generated and is represented in terms of an infinite series of Fourier spatial modes. During the longer piston time period tp* = O(10−2 s) the piston moves at constant speed. A multiple-timescale formulation is used to separate the instantaneous acoustic field from the accumulated bulk response of the gas to piston compression. The latter is found to be identical to the classical quasi-static results from equilibrium thermodynamic calculations. Nonlinear effects become important on the piston timescale. Modal interactions are represented by a system of coupled, nonlinear ordinary differential equations for the time-dependent Fourier coefficients. A numerical solution for this system describes the wavefront steepening to form a weak shock and its propagation back and forth repeatedly inside the cylinder.

Two-photon excitation of NaI with femtosecond laser pulses

We have performed a quantum analysis of the two-pulse excitation of NaI, pertinent to the experiments of Rosker, Rose, and Zewail. The model for the NaI molecule is based on early work by Faist, Levine, Grice, and Herschbach. The quantum calculation uses a time-dependent fast Fourier transform method and is exact. We examine the dependence of the laser-induced fluorescence (LIF) signal on the properties of pump and probe pulses and on the dynamics of the excited molecule. Several important differences between the LIF method with ultrashort pulses and the cw case are pointed out.

The evolution to steady nonlinear pulsation in stellar models

The numerically generated dynamical evolution of an RR Lyrae model from different initial conditions is subjected to a time-dependent Fourier analysis, which yields the temporal behavior of the amplitudes and phases of the few longlived transient modes in addition to the ultimate winner. It is shown that the amplitude equation formalism of Buchler and Goupil gives an almost perfect fit to the observed transient behavior of the amplitudes and phases of the excited modes. Prospects and applications are discussed.

Rotating discontinuity of the bowed string using computer graphics

Starting from Helmholtz’ observations on the time dependence of the motion of a point on a bowed string, the Fourier series for this experimental curve is obtained. By comparing this to the general time-dependent Fourier series for a string fixed at both ends, a general expression for the displacement of the bowed string as a function of position and time is derived. Using computer graphics facilities this function is plotted versus position 12 times during a period. The same is done for the velocity on the string. Finally, the velocity as a function of time for five different positions on the string is plotted.

Application of time-dependent Fourier analysis to nonlinear pulsational stellar models

Application of time-dependent Fourier analysis to nonlinear pulsational stellar models

Modal selection in stellar pulsators. II - Application to RR Lyrae models

The numerically produced dynamical evolution of several RR Lyrae models from various initial conditions is subjected to a time-dependent Fourier analysis. This generates the temporal behavior of the amplitudes and phases of the few long-lived transient modes in addition to those of the ultimately surviving mode. It is shown that the amplitude equation formalism of Bucher and Goupil (1985) gives a remarkably good description of the observed behavior of the models. It is concluded (1) that the essence of the dynamical behavior of the model is played in a low-dimensional subspace of phase space, (2) that the nonlinearities in the pulsation are so weak as to involve only the lowest essential nonlinearities, and (3) that the assumptions made by Buchler and Kovacs (1986) in their discussion of the modal selection problem are satisfied. 23 references.

Switching between dynamic states in intermediate-length Josephson junctions.

The appearance of zero-field steps (ZFS's) in the current-voltage characteristics of intermediate-length overlap-geometry Josephson tunnel junctions described by a perturbed sine-Gordon equation (PSGE) is associated with the growth of parametrically excited instabilities of the McCumber background curve (MCB). A linear stability analysis of a McCumber solution of the PSGE in the asymptotic linear region of the MCB and in the absence of magnetic field yields a Hill's equation which predicts how the number, locations, and widths of the instability regions depend on the junction parameters. A numerical integration of the PSGE in terms of truncated series of time-dependent Fourier spatial modes verifies that the parametrically excited instabilities of the MCB evolve into the fluxon oscillations characteristic of the ZFS's. An approximate analysis of the Fourier mode equations in the presence of a small magnetic field yields a field-dependent Hill's equation which predicts that the major effect of such a field is to reduce the widths of the instability regions. Experimental measurements on Nb-${\mathrm{Nb}}_{\mathrm{x}}$${\mathrm{O}}_{\mathrm{y}}$-Pb junctions of intermediate length, performed at different operating temperatures in order to vary the junction parameters and for various magnetic field values, verify the physical existence of switching from the MCB to the ZFS's. Good qualitative, and in many cases quantitative, agreement between analytic, numerical, and experimental results is obtained.

SEM approach to transient scattering by a lossy, radially inhomogeneous dielectric circular cylinder

The singularity expansion method is applied to the computation of fields associated with the transient scattering of a pulsed electromagnetic plane wave of finite duration by an isotropic, lossy, radially inhomogeneous dielectric circular cylinder. An angular Fourier transformation reduces the problem to solving a series of one-dimensional transient scattering problems, each of which is similar to the problem of scattering by a dielectric slab. Each of the resulting time-dependent Fourier coefficients is then expressed as a Laplace inversion integral over the Bromwich contour. This integral is evaluated by supplementing the contour of integration by circular arcs at infinity, and applying Cauchy's theorem. For each angular order, this leads to contributions from a denumerable set of poles and from a branch cut along the negative half of the real-frequency axis. Procedures are devised to determine these singularities and their contributions numerically. For a representative configuration, results are presented and discussed. A physical interpretation of these results is given.

A quantum-mechanical time-dependent simulation of the scattering from a stepped surface

A recently developed time-dependent quantum-mechanical Fourier method is employed for the simulation of scattering atoms from surfaces of solids. The method allows exact calculation of scattering intensities, resonance strengths and lifetimes, in systems with a very large number of populated diffraction peaks. The study applies the method to the He/stepped copper surface. The accuracy of the method is checked by comparing to close coupling results for the He/W system.

Instantaneous power spectra

Conventional spectral analysis methods do not describe the distribution of signal power in both time and frequency simultaneously. Time dependent Fourier transforms are defined to extend the conventional theory to include instantaneous energy and power spectra for the past and future signal. These spectra are combined to define an instantaneous power spectrum (IPS) of the entire signal. The IPS is a noncausal mathematical property of the signal from which observable properties can be derived, e.g., the lagged window Fourier transform estimate of the time‐frequency variations of the signal. The IPS for a nonstationary process is shown to be given by the Fourier transform of Loeve’s generalized power spectrum of the process and to be a generalization of the Wiener–Kintchine theorem. It is also shown to represent the special cases of the stationary process, the locally stationary process and the deterministic process.

Thermal analysis of three zone solar pond

This paper presents a periodic analysis of a three zone solar pond as a solar energy collector and long term storage system. We explicitly take into account the convective heat and mass flux through the pond surface and evaluate the temperature and heat fluxes at various levels in the pond during its year round operation by solving the time dependent Fourier heat conduction equation with internal heat generation resulting from the absorption of solar radiation in the pond water. Eventually, an expression, for the transient rate at which heat can be retrieved from the solar pond to keep the temperature of the zone of heat extraction as constant, is derived. Heat retrieval efficiencies of 40.0 per cent, 32.1 per cent, 28.3 per cent and 25.5 per cent are predicted at collection temperatures of 40, 60, 80 and 100°C, respectively. the retrieved heat flux exhibits a phase difference of about 30 to 45 days with the incident solar flux; the load levelling in the retrieved heat flux improves as the thickness of the non-convective zone increases. the efficiency of the solar pond system for conversion of solar energy into mechanical work is also studied. This efficiency is found to increase with collection temperature and it tends to level around 5 per cent at collection temperatures about 90°C.

Space-Time Dynamics of a Fast Breeder Reactor for Localized Disturbances

A method of analysis is developed for nuclear reactor accident initiating events that are localized in space. The method is based on a flux factorization technique, accounting for the flux shape changes taking place near the region of perturbation. In the steady state, the neutron shape functions are expanded in a series of eigenfunctions of the steady-state group removal operator. During the unsteady state, the time-dependent group shape functions are expanded in a series of the same stationary eigenfunctions with time-dependent Fourier coefficients. An auxiliary function is added to this expansion to take account of the spatial variation of the spectral hardening of neutrons in the immediate vicinity of the disturbed region. From the resulting representation of the group shape functions, the equations to be satisfied by the time-dependent Fourier coefficients and the time-dependent auxiliary shape function due to the disturbed region are developed consistently. A typical large [1000-MW(e)] liquid-metal fast breeder reactor with two radial core zones of different enrichments is analyzed by the above method. The transient initiating perturbation is taken to be a specified rate of coolant voiding from a single subassembly in the reactor core. The results show a strong dependence of the reactivity added on the radial location of the voiding perturbation and on the rate of voiding.

Frequency-dependent diffraction from enzymatic breathing modes.

The intensity of the time-dependent Fourier transform of any macro-molecule which contains a cleft of the sort that exists in the protein lysozyme is derived. It is shown that there is a region of this transform in which this intensity varies nearly sinusoidally with time with a frequency equal to that of the macromolecule's breathing mode of vibration, and where, therefore, the diffracted power will be primarily at this single frequency. Observation of this power will allow the experimental determination of the frequency of this mode, which is expected to be much lower than those observable by Raman spectroscopy.

Propagation of a Longitudinal Disturbance on a One-Dimensional Lattice

Propagation of a particular longitudinal disturbance on a semi-infinite one-dimensional lattice of identical mass particles coupled by identical massless ideal springs is studied. With all but one of the particles at rest and all the connecting springs at equilibrium length at t = 0, the end particle is constrained to move with fixed longitudinal velocity for all t ≥ 0. Use is made of a method in which an infinite system of differential equations of motion is replaced by a single ordinary differential equation; the time-dependent Fourier coefficients of the equation's solution are the respective displacements from equilibrium of the individual lattice particles. These displacements and the corresponding velocities are expressed in terms of definite integrals. The results of velocity computations are presented along with asymptotic velocity distributions on the lattice and are discussed in relation to a simplified propagation description posited in a recent introductory textbook. The case of a semi-infinite elastic continuum subject to essentially the same initial and boundary conditions is worked out as a limit of the solution of the problem for the lattice.